First of all define what you understand for cultural impact.
I think it's pretty self-explanatory. In the world there are cultures, and Gaga has certainly had an impact on several of them, including the LGBT culture, the pop culture in general, as well as academic culture (I believe at least one university has a Lady Gaga course).
Second the only song that have a good critical acclaim is "Poker Face", and maybe "Bad Romance" but we have to wait until the EOY list come out.
Just Dance, Paparazzi and Telephone have all received critical acclaim that is not significantly different from the acclaim of either of those songs. Check out the links to reviews you can find on their Wikipedia pages.
And third, I do not seem that Lady Gaga is promoting something new, just listen to Britney Spears.
"Promoting something new" isn't part of the criteria. That said, Britney Spears is not the same as Lady Gaga. For one thing, on a purely musical level, Gaga's music makes much more use of European electronic music than Britney Spears' music ever did, which has always been firmly rooted in thoroughly American R&B and pop sensibilities. Beyond this, Britney is largely a traditionally packaged heterosexual sex object of a pop star, while Gaga is almost completely androgynous and very much influenced by and an influence upon LGBT culture.